44edf: Difference between revisions

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Created page with "'''44EDF''' is the equal division of the just perfect fifth into 44 parts of 15.9535 cents each, corresponding to 75.2185 edo. It is related to the regula..."
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'''44EDF''' is the [[EDF|equal division of the just perfect fifth]] into 44 parts of 15.9535 [[cent|cents]] each, corresponding to 75.2185 [[edo]]. It is related to the regular temperament which tempers out |183 -51 -44> in the 5-limit, which is supported by 301, 376, 677, 1053, 1429, 1730, 2407, and 2783 EDOs.
{{Todo|cleanup|add examples|text=add examples of how music can be made with this tuning, stuff like instruments tuned to it, 4 to 12 note scales within it, etc.}}
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
44edf corresponds to 75.2185[[edo]]. It is related to the [[regular temperament]] which [[tempering out|tempers out]] {{monzo| 183 -51 -44 }} in the [[5-limit]], which is supported by {{EDOs| 301-, 376-, 677-, 1053-, 1429-, 1730-, 2407-, and 2783edo }}.
 
=== Harmonics ===
{{Harmonics in equal|44|3|2|intervals=prime}}
 
== Related regular temperaments ==
===5-limit 677&1053===
Comma: |183 -51 -44>
 
POTE generator: ~|-104 29 25> = 15.9540
 
Mapping: [<1 1 3|, <0 44 -51|]
 
EDOs: {{EDOs|75, 301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3836}}
 
===2.3.5.11 677&1053===
Commas: 184549376/184528125, 38084983750656/38060880859375
 
POTE generator: ~|-104 29 25> = 15.9535
 
Mapping: [<1 1 3 1|, <0 44 -51 185|]
 
EDOs: {{EDOs|301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3084}}
 
===13-limit 677&1053===
Commas: 6656/6655, 184549376/184528125, 1162261467/1161875000
 
POTE generator: ~|-104 29 25> = 15.9540
 
Mapping: [<1 1 3 1 -3|, <0 44 -51 185 504|]
 
EDOs: {{EDOs|677, 1053, 1730, 2407, 3084, 4137}}


==Intervals==
==Intervals==
Line 9: Line 45:
! | comments
! | comments
|-
|-
| | 0
| colspan="2"| 0
| | 0.0000
| | '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
| |  
| |  
Line 20: Line 55:
|-
|-
| | 2
| | 2
| | 31.9070
| | 31.907
| |  
| |  
| |  
| |  
Line 65: Line 100:
|-
|-
| | 11
| | 11
| | 175.4888
| | 175.48875
| | 31/28
| | 31/28
| |  
| |  
Line 111: Line 146:
| | 20
| | 20
| | 319.0705
| | 319.0705
| | 101/84
| |6/5
| |  
| |  
|-
|-
| | 21
| | 21
| | 335.0240
| | 335.024
| |  
| |  
| |  
| |  
Line 125: Line 160:
|-
|-
| | 23
| | 23
| | 366.9310
| | 366.931
| |  
| |  
| |  
| |  
Line 131: Line 166:
| | 24
| | 24
| | 382.8845
| | 382.8845
| | 126/101
| |5/4
| |  
| |  
|-
|-
Line 141: Line 176:
| | 26
| | 26
| | 414.7916
| | 414.7916
| |  
| |14/11
| |  
| |  
|-
|-
| | 27
| | 27
| | 430.7451
| | 430.7451
| |  
| |9/7
| |  
| |  
|-
|-
Line 166: Line 201:
| | 31
| | 31
| | 494.5592
| | 494.5592
| |  
| |4/3
| |  
| |  
|-
|-
Line 175: Line 210:
|-
|-
| | 33
| | 33
| | 526.4663
| | 526.46625
| | 42/31
| | 42/31, 27/20
| |  
| |  
|-
|-
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| | 39
| | 39
| | 622.1874
| | 622.1874
| |  
| |63/44
| |  
| |  
|-
|-
| | 40
| | 40
| | 638.1409
| | 638.1409
| |  
| |81/56
| |  
| |  
|-
|-
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|-
|-
| | 42
| | 42
| | 670.0480
| | 670.048
| |  
| |  
| |  
| |  
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| | 43
| | 43
| | 686.0015
| | 686.0015
| |  
| |40/27
| |  
| |  
|-
|-
| | 44
| | 44
| | 701.9550
| | 701.955
| | '''exact [[3/2]]'''
| | '''exact [[3/2]]'''
| | just perfect fifth
| | just perfect fifth
|-
|45
|717.8985
|243/160
|
|-
|46
|733.862
|
|
|-
|47
|749.8156
|
|
|-
|48
|765.7691
|14/9
|
|-
|49
|781.7226
|11/7
|
|-
|50
|797.6761
|
|
|-
|51
|813.6297
|8/5
|
|-
|52
|829.5832
|
|
|-
|53
|845.5367
|
|
|-
|54
|861.4902
|
|
|-
|55
|877.44375
|5/3
|
|-
|56
|893.3973
|
|
|-
|57
|909.3508
|27/16
|
|-
|58
|925.3043
|
|
|-
|59
|941.2578
|
|
|-
|60
|957.2184
|153/88
|
|-
|61
|973.1649
|7/4
|
|-
|62
|989.1184
|99/56
|
|-
|63
|1005.0719
|243/136
|
|-
|64
|1021.0255
|9/5
|
|-
|65
|1036.979
|
|
|-
|66
|1052.9325
|
|
|-
|67
|1068.886
|13/7
|
|-
|68
|1084.89355
|15/8
|
|-
|69
|1100.7931
|
|
|-
|70
|1116.7466
|
|
|-
|71
|1132.7001
|
|
|-
|72
|1148.6536
|
|
|-
|73
|1164.9072
|
|
|-
|74
|1180.5607
|160/81
|
|-
|75
|1196.5142
|2/1
|
|-
|76
|1212.4677
|
|
|-
|77
|1228.42125
|
|
|-
|78
|1244.3748
|
|
|-
|79
|1260.3283
|
|
|-
|80
|1276.2818
|
|
|-
|81
|1292.2353
|
|
|-
|82
|1308.1889
|
|
|-
|83
|1324.1424
|
|
|-
|84
|1340.0959
|
|
|-
|85
|1356.0494
|
|
|-
|86
|1372.003
|
|
|-
|87
|1387.9565
|20/9
|
|-
|88
|1403.91
|'''exact''' 9/4
|
|}
|}
==Related regular temperaments==
===5-limit 677&1053===
Comma: |183 -51 -44>
POTE generator: ~|-104 29 25> = 15.9540
Map: [<1 1 3|, <0 44 -51|]
EDOs: 75, 301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3836
===2.3.5.11 677&1053===
Commas: 184549376/184528125, 38084983750656/38060880859375
POTE generator: ~|-104 29 25> = 15.9535
Map: [<1 1 3 1|, <0 44 -51 185|]
EDOs: 301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3084
===13-limit 677&1053===
Commas: 6656/6655, 184549376/184528125, 1162261467/1161875000
POTE generator: ~|-104 29 25> = 15.9540
Map: [<1 1 3 1 -3|, <0 44 -51 185 504|]
EDOs: 677, 1053, 1730, 2407, 3084, 4137
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 17:21, 17 January 2025

← 43edf 44edf 45edf →
Prime factorization 22 × 11
Step size 15.9535 ¢ 
Octave 75\44edf (1196.51 ¢)
Twelfth 119\44edf (1898.47 ¢)
Consistency limit 4
Distinct consistency limit 4

44 equal divisions of the perfect fifth (abbreviated 44edf or 44ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 44 equal parts of about 16 ¢ each. Each step represents a frequency ratio of (3/2)1/44, or the 44th root of 3/2.

Theory

44edf corresponds to 75.2185edo. It is related to the regular temperament which tempers out [183 -51 -44⟩ in the 5-limit, which is supported by 301-, 376-, 677-, 1053-, 1429-, 1730-, 2407-, and 2783edo.

Harmonics

Approximation of prime harmonics in 44edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.49 -3.49 +5.55 -2.63 -3.40 -5.45 -7.22 +7.61 -4.08 -6.54 +5.63
Relative (%) -21.8 -21.8 +34.8 -16.5 -21.3 -34.2 -45.3 +47.7 -25.6 -41.0 +35.3
Steps
(reduced) 		75
(31) 		119
(31) 		175
(43) 		211
(35) 		260
(40) 		278
(14) 		307
(43) 		320
(12) 		340
(32) 		365
(13) 		373
(21)

Related regular temperaments

5-limit 677&1053

Comma: |183 -51 -44>

POTE generator: ~|-104 29 25> = 15.9540

Mapping: [<1 1 3|, <0 44 -51|]

EDOs: 75, 301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3836

2.3.5.11 677&1053

Commas: 184549376/184528125, 38084983750656/38060880859375

POTE generator: ~|-104 29 25> = 15.9535

Mapping: [<1 1 3 1|, <0 44 -51 185|]

EDOs: 301, 376, 677, 978, 1053, 1429, 1730, 2407, 2783, 3084

13-limit 677&1053

Commas: 6656/6655, 184549376/184528125, 1162261467/1161875000

POTE generator: ~|-104 29 25> = 15.9540

Mapping: [<1 1 3 1 -3|, <0 44 -51 185 504|]

EDOs: 677, 1053, 1730, 2407, 3084, 4137

Intervals

degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 15.9535
2 31.907
3 47.8606
4 63.8141
5 79.7676 22/21
6 95.7211 37/35
7 111.6747 16/15
8 127.6282 14/13
9 143.5817 25/23
10 159.5352 34/31
11 175.48875 31/28
12 191.4423 19/17
13 207.3958 62/55
14 223.3493 33/29
15 239.3028 31/27
16 255.2564 51/44
17 271.2099 62/53
18 287.1634
19 303.1169 81/68
20 319.0705 6/5
21 335.024
22 350.9775 60/49, 49/40
23 366.931
24 382.8845 5/4
25 398.8381 34/27
26 414.7916 14/11
27 430.7451 9/7
28 446.6986 22/17
29 462.6522
30 478.6057
31 494.5592 4/3
32 510.5127
33 526.46625 42/31, 27/20
34 542.4198
35 558.3733
36 574.3268
37 590.2803 45/32
38 606.2339
39 622.1874 63/44
40 638.1409 81/56
41 654.0944
42 670.048
43 686.0015 40/27
44 701.955 exact 3/2 just perfect fifth
45 717.8985 243/160
46 733.862
47 749.8156
48 765.7691 14/9
49 781.7226 11/7
50 797.6761
51 813.6297 8/5
52 829.5832
53 845.5367
54 861.4902
55 877.44375 5/3
56 893.3973
57 909.3508 27/16
58 925.3043
59 941.2578
60 957.2184 153/88
61 973.1649 7/4
62 989.1184 99/56
63 1005.0719 243/136
64 1021.0255 9/5
65 1036.979
66 1052.9325
67 1068.886 13/7
68 1084.89355 15/8
69 1100.7931
70 1116.7466
71 1132.7001
72 1148.6536
73 1164.9072
74 1180.5607 160/81
75 1196.5142 2/1
76 1212.4677
77 1228.42125
78 1244.3748
79 1260.3283
80 1276.2818
81 1292.2353
82 1308.1889
83 1324.1424
84 1340.0959
85 1356.0494
86 1372.003
87 1387.9565 20/9
88 1403.91 exact 9/4
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